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Phase space conceptsY. Papaphilippou, N. Catalan Lasheras

USPAS, Cornell University, Ithaca, NY20th June – 1st July 2005

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OutlineTransverse phase spaceBeam representationBeam EmittanceLiouville theoremNormalized emittanceBeam matrixRMS emittanceTwiss functions revisitedMatched beamEmittance conservation, growth and filamentationBeam distributions

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The equation of the ellipse is

with α,β,γ, the twiss parametersDue to large number of particles, need of a statistical description of the beam, and its size

Under linear forces, any particle moves on ellipse in phase space (x,x’), (y,y’).Ellipse rotates and moves between magnets, but its area is preserved.The area of the ellipse defines the emittance

Transverse Phase Space

x

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Beam is a set of millions/billions of particles (N)A macro-particle representation models beam as a set of n particles with n

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Liouville emittance

Emittance represents the phase-space volume occupied by the beamThe phase space can have different dimensions

2D (x, x’) or (y, y’) or (φ, Ε)4D (x, x’,y, y’) or (x, x’, φ, Ε) or (y, y’, φ, Ε)6D (x, x’, y, y’, φ, Ε)

The resolution of my beam observation is very large compared to the average distance between particles.The beam modeled by phase space distribution function

The volume of this function on phase space is the beam Liouville emittance

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The evolution of the distribution function is described by Vlasovequation

Mathematical representation of Liouville theorem stating the conservation of phase space volume In the presence of fluctuations (radiation, collisions, etc.) distribution function evolution described by Boltzmann equation

The distribution evolves towards a Maxwell-Boltzmann statistical equilibrium

Vlasov and Boltzmann equations

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2D and normalized emittanceWhen motion is uncoupled, Vlasov equation still holds for each plane individually

The Liouville emittance in the 2D phase space is still conservedIn the case of acceleration, the emittance is conserved in the but not in the (adiabatic damping)Considering that

the beam is conserved in the phase spaceDefine a normalised emittance which is conserved during acceleration

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We would like to determine the transformation of the beam enclosed by an ellipse through the acceleratorConsider a vector u = (x,x’,y,y’,…) in a generalized n-dimensional phase space. In that case the ellipse transformation is

Application to one dimension gives and comparing with provides the beam matrix which can be expanded to more dimensions Evolution of the n-dimensional phase space from position 1 to position 2, through transport matrix

Beam matrix

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The average of a function on the beam distribution defined

Taking the square root, the following Root Mean Square (RMS)quantities are defined

RMS beam size

RMS beam divergence

RMS coupling

Root Mean Square (RMS) beam parameters

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RMS emittanceBeam modelled as macro-particlesInvolved in processed linked to the statistical sizeThe rms emittance is defined as

It is a statistical quantity giving information about the minimum beam sizeFor linear forces the rms emittance is conserved in the case of linear forcesThe determinant of the rms beam matrixIncluding acceleration, the determinant of 6D transport matrices is not equal to 1 but

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Beam Twiss functionsThe best ellipse fitting the beam distribution is

The beam Twiss functions can be defined through the rms emittance

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The accelerator itself define ideal betatron functions for motion of individual particles on ellipses In order for the beam to be matched to the accelerator, the beam density should be constant on these ellipseIn other words, the beam Twiss functions should be equal to the betatron functions imposed by the machine structure

Matched beam

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matched mismatched

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RMS emittance growth through filamentation

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When non-linear fields are present, particles in different positions in phase space have different frequency When the beam is rms mismatched, this gives filamentation leading to rms emittance growth

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Kapchinsky-Vladimirsky (KV) distributionK-V is a continuous beam whose distribution projection is uniform in 2D sub phase-spaces

for whichThe beam boundary is

Waterbag distributionSpatial density decreases with radius but distribution is uniform in 4D phase space

The beam boundary is

Linear beam distributions

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The Gaussian distribution has a gaussian density profile in phase space

for which The beam boundary is

KV 4D Waterbag 6D Waterbag Gaussian

Gaussian distribution

Phase space concepts��Y. Papaphilippou, N. Catalan LasherasOutlineTransverse Phase SpaceBeam representationLiouville emittanceVlasov and Boltzmann equations 2D and normalized emittanceBeam matrixRoot Mean Square (RMS) beam parametersRMS emittanceBeam Twiss functionsMatched beamRMS emittance growth through filamentationLinear beam distributionsGaussian distribution